PROJECT SUMMARY Technological development in many emerging research fields has provided us with large collections of data of extraordinary complexity. Brain imaging technology, for example, can generate complex collections of signals from individuals in different neurophysiological states or clinical conditions. Developing new statistical tools to analyze these rich data sets has become a limiting factor for the advancement of medical diagnosis and biomedical research. The goal of this research proposal is to develop new nonparametric and robust methods for analyzing general functional data with complicated structure, such as images, using the idea of depth. In the last two decades there has been an intensive development of notions of data depth, which have become powerful nonparametric tools for analyzing multivariate and functional data. The methods proposed in this project are based on a notion of data depth for general functions and the sample rank-order it provides. Robust nonparametric statistics are particularly relevant in this setting since usually few assumptions can be made about the data generating process and potential outliers, which may be very difficult to detect, can affect the analysis in many different ways. A taxonomy of the different possible types of outliers and exploratory/visualization tools for detecting them will be developed. New approaches based on novel envelope tests for checking if different groups of functions or images come from the same distribution are proposed and will be studied. Recently, the PI has started collaborating with investigators at New York State Psychiatric Institute, led by Dr. Todd Ogden, on a data set that consists of positron emission tomography (PET) brain images from a sample of individuals with major depressive disorders and a sample of controls. The PI has also been working with Dr. Vidhu Thaker, a pediatrician at Columbia University, on analyzing body mass index (BMI) trajectories of children with different degrees of severe early childhood obesity. The methods introduced in this project will extract from these data sets information of clinical relevance far beyond what has been accomplished so far. In particular, the proposed depth-based nonparametric methods will be used to: 1) rank a sample of functions from center-outwards, 2) identify outliers in the data set and 3) develop nonparametric envelope tests for groups differences and identify patterns. We believe that this work will boost the progress in different areas of biomedicine.